Accurate migration of 3D seismic data enables proper interpretation of subterranean hydrocarbon reservoirs. Seismic migration is essentially the process of reversing seismic wave propagation; therefore much effort has been put into modeling seismic wave propagation as accurately as possible. One method of modeling seismic wave propagation is finite difference modeling.
In finite difference modeling, the solution to the wave equation is approximated using a finite difference (FD) method. This method can produce multiple approximate solutions that are referred to according to their order, for example second-order FD and fourth-order FD, which indicates how accurately they represent the true solution to the wave equation. When a second-order FD solution is used to model seismic wave propagation, the resultant synthetic seismic data has significant temporal dispersion due to the low accuracy of the FD solution. Higher order FD solutions such as fourth-order produce synthetic seismic data with less temporal dispersion. The improved accuracy of higher order solutions comes at a great computational cost. For example, fourth-order FD seismic modeling requires twice as many computing operations as second-order FD.
Another method that can be used to improve modeling accuracy is a pseudo-analytic operator such as a pseudo-Laplacian. This method is similar to second-order FD modeling except it modifies the spatial and temporal derivatives so that they have opposite signs and, by adjusting the coefficients, the errors in the derivatives will counteract each other, thereby reducing the inaccuracy in the result. This method is more accurate than second-order FD modeling and not as computationally expensive as fourth-order FD modeling. However, pseudo-analytic methods are still more expensive than second-order FD modeling.
The computational cost associated with using higher order FD modeling or pseudo-analytic methods becomes even more significant when considered in terms of reverse time migration (RTM). In reverse time migration, a source wavefield is propagated forward into the subsurface, often using FD or pseudo-analytic modeling, while a recorded seismic dataset is propagated backwards into the subsurface. The two wavefields are matched at subsurface locations via an imaging condition, often zero-lag cross-correlation, to create an image. The recorded seismic dataset is the true result of seismic energy that has passed through the subsurface and, as such, does not have the temporal dispersion that arises from the approximated FD or pseudo-analytic modeling. In order for the forward propagated synthetic seismic data to match the backward propagated recorded seismic data, the temporal dispersion must be accounted for. Therefore, current RTM methods use higher order FD or pseudo-analytic modeling.
Current practice for FD seismic modeling and RTM uses pseudo-Laplacian methods or higher order FD methods. These methods are more accurate and more computationally expensive than conventional second-order FD modeling.